What are some common uses of HTML forms that contain just one text input and a submit button?

The only one use I find is to submit search queries.

Is there some other reason why this is commonly used?

It is mostly used for search.
But it can be also used for feedbacks.
Its better to have a single field to avoid confusion on the user side.

  • A password reset form.
  • Signing up for a email newsletter.

If I set around really thinking about it, I could probably think of some more.

pixelsoul your right , it has many reason . It depend on the programmer what he/she will do.